A Note on Some Statistical Properties of Signature Transform Under Stochastic Integrals

Xin Guo, Ruixun Zhang, Chaoyi Zhao

Primary Area: general machine learning (i.e., none of the above)

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Keywords: Signature, Lasso, Consistency, Stochastic Integral, Time Series, Universal Nonlinearity

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TL;DR: We investigate some statistical properties of signature transform under stochastic intergrals with a Lasso regression framework, both theoretically and numerically.

Abstract: Signature transforms are iterated path integrals of continuous and discrete-time time series data, and their universal nonlinearity linearizes the problem of feature selection. This paper revisits some statistical properties of signature transform under stochastic intergrals with a Lasso regression framework, both theoretically and numerically. Our study shows that, for processes and time series that are closer to Brownian motion or random walk with weaker inter-dimensional correlations, the Lasso regression is more consistent for their signatures defined by Itô integrals; for mean reverting processes and time series, their signatures defined by Stratonovich integrals have more consistency in the Lasso regression. Our findings highlight the importance of choosing appropriate definitions of signatures and stochastic models in statistical inference and machine learning.

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Submission Number: 5769